Day 1, Mat385: Welcome and First Thoughts

Next time: section 1.1

• Welcome!

• 4x6 cards:
  1. Name (that you want me to call you)
  2. Hometown
  3. What is your calling? ("a calling has to do with one's larger purpose, personhood, deepest values, and the gift one wishes to give the world.... A calling is about the use one makes of a career." David Orr, Earth in Mind)
  4. What is your dream job?
  5. What is something special about you?

• Syllabus and approximate schedule.

• About the project:
  • Projects will be relatively wide open. You may want to find a partner to work with (at most two -- no groups of three).
  • You should find some aspect of discrete math that you want to pursue (perhaps one that we don't study in detail, perhaps one you are particularly interested in that we DO study).
  • You'll be writing these up to put on a wiki.
  • Start early, finish early! It's due the week before finals.
  • Talk to me about what you'd like to do; I'm sure that together we can make it work. I want your project to be about something that you're interested in....

• Overview of the material of the course

  We will be covering portions of the following chapters:

  • Chapter 1: Logic
    • strip away meaning in order to concentrate on form
    • translate English into the symbolism of logic (formalize what is often left imprecise)
    • logical reasoning cannot help but stand you in good stead
  • Chapter 2: Proofs, Recursion, and Analysis of Algorithms
    • Direct versus indirect proofs, especially for proving logical arguments (from chapter 1).
    • Mathematical induction, leading to recursion.
    • Determining the quality of algorithms, by recursion.
  • Chapter 3: Sets
    • Elementary aspects of set theory
    • Counting elements of sets: why some infinities are bigger than others.
    • The similarity between logic and set theory will be more apparent later
  • Chapter 5: Graphs and Trees
    • Graphs and Trees capture information in many different contexts (e.g. a family tree, a unix file system)
    • How can we manipulate, store, and access information stored in a graph?
  • Chapter 6: Graph algorithms
    • shortest path, minimal spanning tree, traversals, etc.
    • Constructing optimal graphs
  • Chapter 7: Boolean Algebras and Computer Logic
    • Operations in logic and set theory are specific cases of Boolean Algebras
    • There is a nice correspondence between boolean expressions, truth functions (or tables), and logic networks (which carry out logical functions for technological applications)
    • We simplify, or optimize, logic networks
    • Binary arithmetic is a particular special application of this section
  • Chapter 8: Finite State Machines
    • Regular expressions
    • We build machines that recognize expressions (strings of characters), and optimize the machines.

• For Wednesday: read Section 1.1; Get an early start on the homework (see the schedule).

• Because of scenes like this:
  and this:
  

  I have instituted the "Class Agreement".

  

• What is discrete mathematics? What does the modifier "discrete" get us? What other kinds of mathematics are there?

  I thought that we'd explore an ancient example of "discrete mathematics", yet one which is tied intricately with the mathematics of calculus and differential equations. Your job: explain it, and discover the secrets contained within it....

  It's called "Yanghui's triangle".

  It has a more familiar name (and if you recognize it, don't shout it out!); but more on that in a bit. For the moment, just look for patterns.

  What can you learn about this from its structure alone?

  • What is this thing? (It has applications to many branches of mathematics.)
  • Your job: identify every pattern you can!
  • What can you learn about the system of writing?

• Next time: Section 1.1